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Can this differential equation be solved using separation of variables?

(dy)/(dx)=2y-xy
Choose 1 answer:
(A) Yes
(B) No

Can this differential equation be solved using separation of variables? $\newline$ $\frac{d y}{d x}=2 y-x y$ $\newline$ Choose $1$ answer: $\newline$ (A) Yes $\newline$ (B) No

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Intermediate Value Theorem

Full solution

Q. Can this differential equation be solved using separation of variables?

\newline

\frac{d y}{d x}=2 y-x y

\newline

Choose

1

answer:

\newline

(A) Yes

\newline

(B) No

Check for Separation: To determine if the differential equation can be solved using separation of variables, we need to see if we can express the equation in the form of $(\frac{dy}{dx}) = g(x)dx$ , where $g(x)$ is a function of $x$ only.
Rewrite the Equation: First, we rewrite the differential equation as $\frac{dy}{dx} = y(2 - x)$ .
Separate Variables: Now, we attempt to separate the variables by dividing both sides of the equation by $y$ and multiplying both sides by $dx$ to get $(1/y)dy = (2 - x)dx$ .
Integrate Both Sides: We have successfully separated the variables, with $y$ on one side and $x$ on the other. This means we can integrate both sides with respect to their respective variables.
Final Solution: The integral of $(1/y)\,dy$ is $\ln|y|$ , and the integral of $(2 - x)\,dx$ is $2x - (x^2)/2 + C$ , where $C$ is the constant of integration.
Final Solution: The integral of $(1/y)\,dy$ is $\ln|y|$ , and the integral of $(2 - x)\,dx$ is $2x - (x^2)/2 + C$ , where $C$ is the constant of integration.Thus, the solution to the differential equation after integrating both sides is $\ln|y| = 2x - (x^2)/2 + C$ .

More problems from Intermediate Value Theorem

Question

f

be a continuous function on the closed interval

[1,5]

f(1)=1

f(5)=-3

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

f(c)=2

for at least one

c

-3

1

\newline

f(c)=-2

for at least one

c

-3

1

\newline

f(c)=2

for at least one

c

1

5

\newline

f(c)=-2

for at least one

c

1

5

Posted 3 months ago

Question

h

be a continuous function on the closed interval

[-3,4]

h(-3)=-1

h(4)=2

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

h(c)=1

for at least one

c

-3

4

\newline

h(c)=-2

for at least one

c

-3

4

\newline

h(c)=-2

for at least one

c

-1

2

\newline

h(c)=1

for at least one

c

-1

2

Posted 3 months ago

Question

g

be a continuous function on the closed interval

[-3,3]

g(-3)=0

g(3)=6

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

g(c)=-2

for at least one

c

-3

3

\newline

g(c)=-2

for at least one

c

0

6

\newline

g(c)=5

for at least one

c

-3

3

\newline

g(c)=5

for at least one

c

0

6

Posted 3 months ago

Question

g

be a continuous function on the closed interval

[-1,4]

g(-1)=-4

g(4)=1

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

g(c)=3

for at least one

c

-1

4

\newline

g(c)=-3

for at least one

c

-4

1

\newline

g(c)=3

for at least one

c

-4

1

\newline

g(c)=-3

for at least one

c

-1

4

Posted 3 months ago

Question

g

be a continuous function on the closed interval

[-1,3]

g(-1)=-2

g(3)=-5

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

g(c)=-3

for at least one

c

-5

-2

\newline

g(c)=0

for at least one

c

-5

-2

\newline

g(c)=0

for at least one

c

-1

3

\newline

g(c)=-3

for at least one

c

-1

3

Posted 3 months ago

Question

f

be a continuous function on the closed interval

[-2,1]

f(-2)=3

f(1)=6

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

f(c)=0

for at least one

c

-2

1

\newline

f(c)=0

for at least one

c

3

6

\newline

f(c)=4

for at least one

c

-2

1

\newline

f(c)=4

for at least one

c

3

6

Posted 3 months ago

Question

f

be a continuous function on the closed interval

[-5,0]

f(-5)=0

f(0)=5

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

f(c)=-2

for at least one

c

-5

0

\newline

f(c)=2

for at least one

c

0

5

\newline

f(c)=2

for at least one

c

-5

0

\newline

f(c)=-2

for at least one

c

0

5

Posted 3 months ago

Question

f

be a continuous function on the closed interval

[1,5]

f(1)=1

f(5)=-3

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

f(c)=2

for at least one

c

1

5

\newline

f(c)=-2

for at least one

c

-3

1

\newline

f(c)=-2

for at least one

c

1

5

\newline

f(c)=2

for at least one

c

-3

1

Posted 3 months ago

Question

h(x)=\frac{1}{x}

\newline

Can we use the intermediate value theorem to say the equation

h(x)=0.5

has a solution where

-1 \leq x \leq 1

\newline

1

\newline

(A) No, since the function is not continuous on that interval.

\newline

0

5

h(-1)

h(1)

\newline

(C) Yes, both conditions for using the intermediate value theorem have been met.

Posted 3 months ago

Question

h

be a continuous function on the closed interval

[0,4]

h(0)=2

h(4)=-2

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

h(c)=3

for at least one

c

0

4

\newline

h(c)=-1

for at least one

c

0

4

\newline

h(c)=-1

for at least one

c

-2

2

\newline

h(c)=3

for at least one

c

-2

2

Posted 3 months ago